Ways to Improve Problem Solving Skills and Math Communication
Do your middle school math students struggle with problem solving? Do they get to the end of the word problem and then guess at the operation they need to choose (maybe not realizing that there are multiple operations)? You probably see this with some of your students, while other students do very well with problem solving. What methods have you found to help those who struggle? What methods can you use to help each student at his or her current level?
I’ve used many strategies over the years, to help students sort out how to make sense of word problems and how to approach them. These methods didn't have a specific name at the time (like close reading or talking to the text), but some would fit into these categories.
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Use Task Cards in a New Way, to Provide
SelfDifferentiation and Promote Discussion
If you're like me (and so many other teachers), you know that task cards can be used in sooo many ways. From centers to Footloose (or Scoot) to exit tickets to entrance tickets to miniquizzes  the list is long!
However, if you're like me in other ways, you're always looking for something new and different. This year, my "new and different" was to start using task cards to play Truth or Dare in math and language arts classes! To use them this way, some of the task/question cards need to be written as True or False questions, which can make the questions just a little trickier and lead to more indepth thinking. I allow students to discuss the answers after the "official" answer is given, and depending on the question, students end up having great discussions! The Dare questions are a little harder, require more calculation or perhaps more verbal explanation than the Truth cards, and so they are worth more points. (Truth cards are worth one point while Dare cards are worth 2 or 3  I've even thrown in a 4pointer here and there.) What makes this game fun? Well, it's a little different  with the "dare" part in there. Students also don't always know how many points they're going to get to try, so that offers a little excitement. I like the fact that students can choose the type of question they want, so it allows for some selfdetermined differentiation...the choice gives the more hesitant students the chance to feel a little more confident. After creating several paper and pencil Truth or Dare games, my wonderful friend Leah (Secondary Resources for Social Studies & English) suggested that I make a Google classroom version, and I'm so glad I did! It's so easy to use and there's little to no copying needed! (A little copying if I want students to write their work/answers on paper; no copying if I want to share the Truth or Dare game in Edit mode and have students type their answers.) Check out the 2minute video below  it shows how the game works in Edit mode (there are one or two "slow to refresh" spots in the video, so please don't think it's not working:)
Check out this video to learn more about the way the game is played with paper/pencil  in any subject!
I hope you can use this game ideait can be used in any subject!
Have I mentioned that I love Jo Boaler’s books and site, Youcubed.org? Well, I do! She shares so much fantastic research and so many wonderful ideas.
So, I was reading her book Mathematical Mindsets this week, and read about the “array game” (called How Close to 100), which I’ve seen all over Pinterest and thought was very cool. I tried it with my classes last year during a little bit of down time, and they liked it. I hadn't really thought of using it this year, but last week I noticed the baggie of polyhedral dice that I've had for a looooong time and thought it would be cool to use the dodecahedron dice for the array game. With these dice, the students could use numbers up to 12, rather than 6.
To set up their game, students each outlined a 20 by 20 area on their own graph paper. They took turns rolling their dice and creating arrays to represent the multiplication problem they had rolled. It was very interesting to observe the way students arranged their arrays. Some started in the corner and worked their way out, while others started on one side and worked their way across. Some made the arrays touch, if possible, while others left a row between each one. Some just drew their first few arrays anywhere and then discovered that they didn't have a lot of room to fit additional ones. The "winner" was the student with the fewest number of boxes left (some did get to zero left). The students really had fun with this!
Of course, some finished their games earlier than others. In these cases, I asked students to create arrays that used different numbers than the numbers they rolled, but represented the same area. For example, if they rolled 12 and 5, their arrays could be 10 by 6, 15 by 4, or 20 by 3 (not 30 by 2, we discussed, because the grid is only 20 by 20). If they rolled a number that couldn't be represented by a wholenumber array, they could then use an irregular shape, or a triangle  anything they could find the area of. It was interesting to see how some students got stumped when they tried to draw an irregular shape to represent a number like 81.
Most students enjoyed this twist (we continued it the next day so they all got to play this version), but a few complained that it made their heads hurt! That's ok...I know they were really thinking and growing mathematically! The next extension for early finishers (only a few) was to use the icosahedron (20sided) dice, and have students create area models to cover their grids and find the answer to the multiplication problems. This required a larger grid, so I had them tape 2 pieces of graph paper together and create 20 by 40 grids. Using the icosahedron dice gave a mix of 1digit by 1digit, 1 by 2digit, and 2 by 2digit problems to model and solve. Most students didn't get very far with this before we ran out of time, but I think this is a great way to them to visualize what multiplying by a twodigit number means. I'd like to revisit this one! I'm so glad I thought about using those polyhedral dice! Have you used polyhedral dice in your math classroom? If so, please share how!
Although we're trying to get through quite a bit of material before our state testing, we took some time today to explore triangles. I'm sure many of you may have done this exploration, but it was quick and fun, so I thought I'd share:) We explored the idea that the sum of the two smaller sides of a triangle must be greater than the longest side. I cut straws of three different lengths, and asked students (in groups) to use the straws to make a triangle.
In my first math class, I used straws that were cut to 2 inches, 3 inches, and 5 inches. These lengths, using straws, made it almost possible to make a triangle, even though it shouldn't have been possible. So, I had to insist that their straw ends be lined up perfectly. I wanted to use 3, 5 and 2 inches to show that even these dimensions won't make a triangle, because the sum is equal to the longest side, not longer than it. So, after understanding how precise they had to be and that they couldn't leave segment parts sticking out of the end of the triangle, they came to the conclusion that it couldn't be done. Next I gave the groups a new set of straws that were cut to 3 inches, 3 inches, 5 inches. In this case, they were excited to make their triangles in about 30 seconds! We then discussed why the 3, 2, 5 didn't work and worked our way to "creating" the rule.
For my next classes, I trimmed the 3 inch straws to 2 inches, so that my next classes would have more difficulty getting the ends to meet. It was so funny to hear their comments  "This doesn't work," "Is this a trick question?" "This is impossible!" And then, their excitement when they made the 3, 3, 5 triangle  "We did it first!"
For my next classes, I trimmed the 3 inch straws to 2 inches, so that my next classes would have more difficulty getting the ends to meet. It was so funny to hear their comments  "This doesn't work," "Is this a trick question?" "This is impossible!" And then, their excitement when they made the 3, 3, 5 triangle  "We did it first!" I think (hope!) that they understood the concept....we'll see tomorrow when we go over their homework:)
Remove One is one of my favorite games! It's a great way to teach probability and the students love it. I've been using it nearly every year since I was introduced to it through a program called the Mathline Middle School Math Project, sponsored by PBS (back in 1997?). I was involved in the program through my graduate studies at Allentown College of Saint Francis DeSales (now DeSales University). Anyway, this year, my student teacher is teaching our probability lessons; so she is the one who taught this lesson.
This is how the lesson works: 1. Students use a piece of paper as their "game board" and number the paper from 122 (or 212) . They then place 15 chips next to the numbers. They are told that they can place one chip next to every number and then place the extras next to any number they want. Or, they can leave some numbers with no chips and put several on others. Usually, they place the chips like those in the picture to the right. 2. Once students have their chips set up, the teacher rolls 2 dice and finds the sum of the numbers that are rolled.
3. If students have a chip next to that sum, the students may remove ONE chip from their paper (thus the name of the game Remove One).
4. Play continues, with the teacher rolling the dice and the students removing one chip each time the corresponding sum is rolled. The "winner" is the student who removes all of the chips first.
Without much class discussion, we play the game a second time. Normally, I just ask them to make some quiet observations to themselves before placing their chips again. Students typically notice that the sums of 6, 7, and 8 are rolled the most often and that 2 and 12 are usually rolled the least often, so they arrange their chips differently.
After the second game, we have a discussion about all of the possible outcomes (sums) one can get when rolling 2 dice. We also discuss how many ways there are to roll each of those outcomes, and what the probability is of rolling each sum. We find this probability in fraction form, and then often convert them to decimals and percents.
After this discussion, we play the game for a third time, and students' "game boards" often look a bit different!
This year, since I was observing rather than teaching, I was better able to hear some of the students' quiet comments to each other... "There's a better chance of getting a seven." "I'm not going to put any on 2, because it still hasn't come up."
When I started discussing this lesson with my student teacher, I searched for the lesson online, just in case it was around, and I found it right away. Click HERE to see the full lesson plan from PBS. Have you played this game? What other probability games do your students enjoy?
As I was thinking about school today, I was thinking about one of our next topics: equivalent expressions.
(CCSS.6.EE.A3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.) Last year, I worked this concept in through the use of my daily math warmups, which brought the idea back time and again, and the students did well with it. This year, even though we will spend more time with direct instruction, I was thinking about other ways to use equivalent expressions, and I thought of using them for partnering cards! We can use them many times throughout the year as quick, random reinforcement. You can download these for free, if you'd like. There are 6 pages, with 3 sets on each page, giving you 18 sets (36 students). The set is in the download twice  once with a background and once without. ** Update: I used these today and they worked well! I printed the set with the background and quickly glued them onto index cards this morning, because I wanted to use them today. I'll laminate them before using again. I hope you can use them!
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AuthorHi, I'm Ellie! My mission here is to support teachers as they work to provide engaging, meaningful experiences for their students. I've been in education for 25 years, teaching all subject areas at both the elementary and middle school levels, and am here to share what I've learned through those years, as well as what I continue to learn. I hope you'll find some ideas or resources here to help you out! Categories
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